# Cancel common factors in symbolic product

Suppose I have the following function

``G[n_, k_] := Product[g[n + i]/g[i], {i, 0, k}] ``

where $$n,k$$ are assumed to be natural numbers and $$g$$ is a nonzero symbolic function. I want to simplify the expression `G[n,k+1]/G[n,k]` and cancel all common terms to get `g[n+k+1]/g[k+1]`. However, `Simplify`, `FullSimplify`, `Expand` and `Cancel` all yield the symbolic (not cancelled) product.